Photomultiplier BNC cable impedance matching

If you want to match the impedances at the input and at the output you could use pieces of transmission line and open circuit stubs between the transistor and the "IN"/"OUT" you are talking about.I always prefer open circuit stubs because I find short circuit difficult to create.You must be given the optimum reflection coefficient of your transistor.Use that as the source reflection coefficient (Since you want to minimize the noise figure) and proceed with TL and stub design.At the output you can maximize the gain.Output does not have much to do with noise.Just use conjugate matching.

Most normal BNC terminated lab equipment uses 50 Ohm. 75 Ohm is mainly used for cable tv and related equipment.

Can you read the number printed on the side of the cables? For example RG58is 50 Ohm but RG59 is 75 ohm.

At low frequencies it does not matter although the amplitude may indicate incorrectly if you terminate in a different impedance from which it was designed. For example if you use a function generator that was designed for 50 ohm and terminate it in 75 ohm the amplitude will 20% higher than intended (e.g. if you set it to 1V p-p output you will actually get 1.2V).

What constitutes low frequency depends upon the length of the cable - if the time delay in the cable is more than say 10% of a cycle of the highest frequency component then you may see ringing or other artifacts. The propagation speed in a cable will be about 65% of the speed of light or about 1.5ns per foot. For your 300MHz which has a wavelength of ~2 feet any impedance mismatch of more than a couple of inches could cause problems.