Can you suspect false carding from the following situation?
Dear Frank has omitted one additional possibility - there is a throw-in working against West when (a) it is actually possible to make the contract and (b) West holds T762 of Clubs. Once the Club Q falls on trick three this play should be visible to Declarer.

Cash three Spade tricks ending in Dummy to leave this for Declarer

S: -
H: 3
D: AT94
C: 5  

S: -
H: K7
D: QJ82
C: -


and the possible West holding to be these:

S: -     or  J     or  J    or  J
H: AQ8       AQ8       AQ8      AQ
D: Kx        KX        K        Kx
C: 6         -         6        6


In all cases West has at most three cashing tricks, and no exit - just lead the last Club to West, pitching a diamond from hand, and wait for your last 2 tricks.

Update #2 - Play details:

Spades should be cashed in the order K; then Q; then A to maintain control if Spades break worse then 4-2. Then the Diamond Q should be pitched from hand on the 4th round of Clubs to unblock; there is no need for more than 2 Diamond tricks and we desire to win both in Dummy.

Update #3 - Odds of both lines of play:

After trick 1 there is only a 11*10 / 20*19 = 11/38 chance that Hearts are split 4-4 (ie that both unplaced Hearts are with East, with 9 and 11 cards unknown in West and East respectively). Combined with the Diamond finesse that gives roughly a 64.5% chance to make the contract.

Testing the Club Q first gives a 50% + 1/3 * 48% = 66% chance to make on a straight doubleton Q, plus the additional chance, as here, of an expert defender attempting a subterfuge that fails on an endplay. 

Update:

Note that an expert West holding the Diamond K, and thus knowing that his partner has at most three HCP, will almost always false card in Clubs so that (a) a false card by his partner in the suit is believable; and (b) if East cannot false card in Clubs then Declarer won't know whom to believe because the signals are inconsistent. It is a mandatory play at that level, and cannot hurt because his partner (a) already knows what to return if he gets in and (b) has nothing to be fooled about. 

The guideline here is that when the defensive values are split decidedly unevenly between the two hands, the weaker hand false cards only out of necessity while the stronger hand will false card frequently. This can sometimes assist Declarer in placing cards, but works to Defenders' advantage more frequently.