The maximum value of the frictional force between the two blocks can be determined by analyzing the forces acting on the system and the conditions for maximum frictional force.

1. Forces acting on the system:
- Weight force (mg): Acting downward on both blocks. The magnitude of this force is the same for both blocks.
- Spring force (F_s): Acting upward on block Q due to the compression of the spring. The magnitude of this force is given by Hooke's Law: F_s = kx, where x is the displacement of block Q from its equilibrium position.
- Normal force (N): Acting perpendicular to the contact surface between the two blocks. The magnitude of this force is equal to the weight force (mg) for each block.
- Frictional force (f): Acting parallel to the contact surface between the two blocks. This is the force we want to determine.

2. Conditions for maximum frictional force:
The maximum frictional force occurs when the two blocks are on the verge of slipping relative to each other, which happens when the force of friction reaches its maximum value. In this case, the maximum frictional force occurs when the force of friction is equal to the static frictional force.

3. Analysis:
- When the two blocks are pulled by a distance A, block Q moves with it due to the spring connection. In this case, the displacement of block Q from its equilibrium position is A.
- The spring force (F_s) is given by F_s = kA.
- The normal force (N) is equal to the weight force (mg) for each block.
- The maximum static frictional force (f_max) is given by f_max = μN, where μ is the coefficient of static friction.

4. Calculation:
- Since the two blocks have the same mass, the normal force (N) is the same for both blocks and is equal to mg.
- Therefore, the maximum static frictional force (f_max) is f_max = μmg.
- The spring force (F_s) is given by F_s = kA.
- For the maximum frictional force to occur, the force of friction (f) should be equal to the maximum static frictional force (f_max). Therefore, f = f_max = μmg.
- From the equation of motion, f = ma, where a is the acceleration of block Q. Since block Q is oscillating without slipping, its acceleration (a) is given by a = ω^2A, where ω is the angular frequency of oscillation.
- Equating f = ma, we get μmg = mω^2A.
- The mass m cancels out, giving μg = ω^2A.
- The angular frequency ω is related to the spring constant k by ω = sqrt(k/m).
- Substituting this relation, we get μg = (k/m)A.
- Finally, rearranging the equation, we get μ = (kA)/(mg), which is the maximum value of the coefficient of static friction.

5. Conclusion:
The maximum value of the frictional force between the two blocks is given by f_max = μN = μmg = (kA)/(mg). Therefore, the correct answer is option (a).