This question doesn't relate to the actual speed of the aircraft, but of the blades themselves. I have heard helicopters had problems of this happening if the rotors spin too fast it can break the sound barrier, so could this happen to a turboprop?
Can it happen: Yes
Does it happen: Generally no
As a matter of the pure physics propeller tips can go supersonic and some times do. There is at least one case of this being done by design on the XF-84H which was built to be one of the fastest propeller planes.
The main issue it faced was the noise generated by its supersonic prop. It is considered one of the loudest aircraft ever made as its prop was supersonic even at idle speeds. Due to the shock waves generated by breaking the sound barrier great care is often taken to ensure that propellers don't go supersonic.
It seems sometimes happens on the Tupolev Tu-95 "Bear" long-range strategic bomber. According to various sources like this (the first result from a Google search), the tip of the propeller goes transonic while turning at full rpm with a decrease in performances and a severe increase in noise.
The Tu-95 has straight blades, like many old turboprop aircrafts:
To partially overcome this problem, modern turboprops use "scimitar" blades, like the one on the C-130J:
In the same way a swept wing is more efficient at high speed in comparison to a straight wing, at high tip speed, a scimitar blade is more efficient than a straight blade.