No. " I believe" covers a multitude of false statements; a person may believe that the Earth is flat and not nearly round. That the belief is demonstrably false does not stop someone genuinely having it. Commonly the belief amounts to no more than what the witness was told by someone, malicious or otherwise, and therefore is hearsay which is either not admissible as evidence of fact or, depending on the proceeding, is admitted but no, or very little weight is attached to it; it may be admitted to explain why someone acted as they did, rather than as proof of the facts asserted in it, or to go to the character of the person saying it, to show that they will freely assert anything without any grounds.
If such statements gave rise to a claim for monetary damages or other redress, the courts would be doing little else but try such actions, since barely a case passes, whether criminal or civil, without such statements.
So the short answer is "No". The satisfaction has to be that the statements are not believed. Only in extreme cases, where the plaintiff (complainant in England) has deliberately brought an action based on falsifying records or other deliberate attempt to pervert public justice, does the matter end in prosecution for perjury.That still doesn't mean that a civil action will be brought following conviction.