Although there are no quarter-Vulcan/Human hybrids in the Trek Movie and TV canon, I think we can make some educated guessed about their abilities;
Based on the fact that there are human telepaths, there seems to be no good reason why a Human/Vulcan hybrid would lose that ability, including the ability to mind-meld.
A variety of races have benefitted from Vulcan teachings on logic and emotional control (Suder in Voyager and Trip in Enterprise) and it's likely that with sufficient training a hybrid could achieve a similar level of impassivity
Although there's no canon explanation for it, Spock (who is half-human) is described as being "much stronger than a human" and potentially as strong as a full-blooded Vulcan (e.g. "three times the strength of a human").
A quarter-Vulcan hybrid might have the same physical prowess or they might not.
It's pretty clear that it's possible for a non-Vulcan to learn the Nerve Pinch. It's been performed by Data, Odo, the EMH, Jean-Luc Picard, Seven-of-Nine and Jonathan Archer.
Moving waaaaaaay down the canon scale, Spock had a quarter-Vulcan son (named Zar) in the EU trek book "Yesterday's Son". The boy demonstrates a woeful lack of emotional clarity, has no knowledge of pon farr, cannot do the nerve pinch but does indicate a strong telepathic ability and an ability to project his emotions as a weapon.