No, electric force between two charges or on any one of the two charges does not depends on the charges present nearby.
According to Coulomb’s Law,
Electric force is given as:
Here, k is a constant and k = = 9× 10⁹ Nm²C^-2, q₁ and q₂ are the two charges and r is the distance between two charges. As we see, the force on one charge depends only on the magnitude of the second charge and vice versa.
Third charge won’t affect the force between first two charges.
The only thing that would affect the force on the charge is the net force due to superposition of all the individual forces.

Hence, the force on a charge due to another charge does not depend on the charges present nearby.