If we can't identify any logical fallacies in an argument it is very likely that the argument is valid/strong/cogent (depending on type)?

Does the absence of fallacies in a argument make very probable that the premises support its conclusion?

Valid/strong/cogent all mean very different things in argumentation, so this question can be answered many different ways depending on the combination. Your other question, "Does the absence of fallacies in a argument make very probable that the premises support its conclusion?" is easier to answer.

If we are talking about a formal, deductive argument, then the lack of formal fallacies does mean we have a valid argument (an argument is valid if the truth of the premises logically guarantees the truth of the conclusion). However, it does not mean the conclusion is true because the premises could be false. For example,

All cats can fly. Jamie is a cat. Jamie can fly. (no fallacies, valid, yet very wrong)

If we are talking about inductive or informal arguments, then I would say that the presence of fallacies makes the argument weaker. Remember that informal fallacies are instances of poor reasoning. There are many fallacies that are common and therefore named, but bad reasoning that is hard to categorize and/or rare so there is no named fallacy for it is considered fallacious reasoning. The relationship between fallacies, premises, and the conclusion is much more ambiguous with informal fallacies. And sometimes, facts can simply be very wrong although the reasoning is decent given the incorrect facts.

Now I should also add that just because you (or someone) can't identify any fallacies in an argument, does not mean there are none; it simply means the person looking did not find any.