In MATLAB ,do the input signals need to be of the same sampling rate for the conv function to work correctly?
What an interesting question! An impulse response sequence, taken on its own, does not have a specific numerical sample rate (measured in Hz) associated with it. If I applied a [1, 0, 0, 0, ...] unit impulse sequence to a 3-point moving average lowpass filter then the filter's impulse response would be the sequence [1/3, 1/3, 1/3, 0, 0, 0, ...].  There's no specific numerical sample rate (measured in Hz) associated with either the input unit sample sequence or the filter's impulse response sequence. If you insist on talking about the somewhat unusual notion of the sample rate of a filter's impulse response I suppose you can say, "The sample rate of a filter's impulse response is always equal to the filter's input data sample rate."

For example, if to my 3-point lowpass moving average filter I apply an input sequence whose sample rate is 1000 samples/second (1 kHz), then the filter's output sequence has a sample rate of 1000 samples/second. On the other hand, if I apply an input sequence whose sample rate is 24000 samples/second (24 kHz) to the same lowpass filter, then the filter's output sequence will have a sample rate of 24000 samples/second. (WARNING: I suggest you never discuss the sample rate of a filter's impulse response during a job interview.)