They can construct various internally-consistent models of what's going on, debate the merits of those models, and check them against the available evidence while doing their best to procure and evaluate new evidence. It's not fundamentally dissimilar from what, say, particle physicists do. The raw materials are different and so the research toolkit necessarily reflects those differences, but serious fields of intellectual inquiry all resemble each other in these fundamental ways.

Since you mentioned Q particularly, I remember the college course in which I learned about it was using Stevan Davies' New Testament Fundamentals, which my college-aged self was able to come to grips with pretty readily. For me that answers the question of "how do they prove that to anyone who's not a specialist" — they can publish a (relatively slim!) undergraduate textbook that incorporates the theory.