Conquian is the earliest known Rummy game in the western world - it has been played at least since the 1880's. Its place of origin is probably Latin America - the composition of the 40 card pack being equivalent to the usual form of the Spanish pack. Clark Williams tells me it is played in Philadelphia, Pennsylvania. It is probably played in other parts of the USA as well, and I have heard that it is also played in Mexico.
There are two players.
The deck consists of the cards A-2-3-4-5-6-7-J-Q-K in the usual suits hearts, diamonds, clubs and spades. It can be made by throwing the 8s, 9s, and 10s out of a standard 52 card pack.
The dealer deals ten cards to each player, one at a time. The remainder of the cards are placed face down on the table to form a stock.
The objective is to be the first to "go out" by melding 11 cards. Melding is placing a valid combination of cards face up on the table in front of you. These face up cards are your meld. The combinations allowed are:
7, 
7, 
7.
4, 5, 6. Note that in this game the ace is always low and the 7 is next to the jack. So when making sequences A-2-3 and 6-7-J are legal but Q-K-A is not allowed. At all times, the meld you have on the table must consist one or more separate valid combinations. A card can never be part of more than one combination at the same time.
Note that a sequence cannot contain more than eight cards. This maximum length is set because you have to meld exactly eleven cards to win. For this you must have at least two combinations. If one of them has the minimum size of three cards, the other cannot have more than eight.
The non-dealer begins the play by turning the top card of the stock face up, and has two options:
The turn to play alternates. Apart from the non-dealer's first turn, you always begin your turn with the opportunity to use the card discarded by your opponent or the card which your opponent turned up and did not use. Your options are:
When melding, you can rearrange your table cards in any way you wish, as long as the cards you have on the table at the end of your turn form separate valid combinations. For example, if you have a meld of 3-4-5-6 on the table, 6 in your hand, and you turn up 6, you can meld three sixes, taking the six from your club sequence, leaving a valid sequence 3-4-5. If a four had turned up and you had a four in your hand, you would not be able to meld fours, because stealing the 4 would not leave a valid sequence.
There is one further important rule. If during your turn the face-up centre card can be added to the meld that you already have on the table, without using extra cards from your hand, your opponent can force you to meld the centre card, and you cannot refuse. This can be a powerful technique for destroying a player's position. Suppose that you have 3-3-3-3 6-7-J-Q on the table and A-A in your hand. All you need to win is an ace, but you opponent might discard the K and force you to meld it. You then have no option but to discard one of your aces, and the only way you can now win is to acquire the 5 and 4. If either of these is already out of the game, it has become impossible for you to win.
Please note that
Play continues in this way until either someone goes out or the stock runs out.
You go out by melding the face up card from the centre of the table (whether discarded or rejected by your opponent or newly turned up by you) along with all the remaining cards in your hand (if any). Your meld on the table will then consist of eleven cards. In this case you have won and your opponent pays you a fixed stake.
If there are no cards remaining in the stock and you do not use the card your opponent discarded or rejected, you are unable to turn up a new card because there are none left. In this case the game is a draw, and some play that the next game is played for a double stake.
Note that although you may meld ten cards and discard your last card, this does not win the game or end the play. If you do this you have to continue playing until you get an eleventh card which you can add to your meld. This eleventh card would have to be either discarded or passed to you by your opponent or turned up by you after rejecting the card from your opponent.
Clark Williams has provided the following advice:
All cards must be played on the table. This is often a very intense money game and strict attention to prevent the appearance of cheating is very important
Defensive Play. If you are dealt a hand with more than 5 or 6 cards which cannot be included in any series or grouping, it is possible to simply fold your hand and let the opponent attempt to play alone by continuing to draw and discard. The effect is equivalent to always rejecting the face-up card that is passed to you. The likelihood is that you possess the cards your opponent would need to win, in which case the hand will be tied. In such a case, you can only lose by playing and indeed, may win the next hand. There are times when five or six hands in a row may be tied in such a manner.
Offensive Play. Players with an excellent memory often count cards - they keep track of all cards played by both players and predict those needed to complete the opponent's hand, deliberately playing a grouping which reduces the opponent's chances of winning. In any case, it is vital to keep track of the cards played, to some extent. The better this is done, the higher the probability of winning. Usually, winning or losing eventually depends on the fall of one or two cards, in the end.
In play, you must be careful to avoid the dreaded Ten Card Hole. In the highly improbable event that you may be able to collect all the Clubs, for instance, it would be impossible to win, because there are only ten cards in any suit.
Rummy-games.com also has rules of two-player Conquian as well as the full text of R.F. Foster's book Cooncan (Conquian) A Game of Cards also called "Rum" (1913) which describes a form of basic Rummy with one or two 52-card packs for 3-5 players but gives Conquian with 40 cards as the recommended game for 2 players.
Rummy.com also has rules for two-player Conquian.