General transfer function for a bandpass filter
No. Whilst a standard second-order bandpass section can be defined in this way ...

\$H(s) = \dfrac{\dfrac{\omega_m}{Q}s}{s^2+\dfrac{\omega_m}{Q}s+\omega_m^2}\$

... it is also possible to have a second-order bandpass filter with the same characteristic frequency and Q but with a different transfer function. This previous question which addresses a filter with a stop-band attenuation of 1 is a case-in-point.

Furthermore, higher-order filters will require more than just these two parameters to define them since there are more coefficients.