In short: no. However, there is one rule that most higher education I've encountered seems to stick to: they really like constitutions and written rules.
A quick google of my own university found a faculty senate constitution of over 30 pages with a great degree of detail and rules, and absolutely no mention of conflict of interest.
From further review, I might propose an idea of why: interest is assumed. Rather than being an organization who seeks impartiality, there is actually a deeply vested interest assumed that all senators - and the senate as a whole - try to forward the goals of the university and the interests of the faculty itself. Using my own reference constitution, early on this line states the purpose of the Senate: "The Senate shall be the representative body of the faculty." In other words, they are out for what is best for the faculty.
As such, in the situation you briefly outline, there isn't a conflict of interest, but it sounds like a case instead of "vested interest" - the faculty, and thus the senate, want pay raises and oppose administration that seeks to oppose or refuse it, and so they do what they can to help achieve this mission. In short, this isn't unethical - it's the definition of politics and the reason why the senate exists.
Much of this is determined, at least in the US, by state law and mandate for public universities. In Wisconsin: The faculty derives its authority from 36.09(4) Wisconsin Statutes, which reads as follows:
FACULTY. The faculty of each institution, subject to the
responsibilities and powers of the board, the president, and the
chancellor of such institution, shall be vested with responsibility
for the immediate governance of such institution and shall actively
participate in institutional policy development. As such, the faculty
shall have the primary responsibility for academic and educational
activities and faculty personnel matters. The faculty of each
institution shall have the right to determine their (sic) own faculty
organizational structure and to select representatives to participate
in institutional governance.
As with all good representative systems, it is assumed that there are checks and balances. There are usually student government wings, faculty senates, administrative positions, committees, etc - all designed to, hopefully, ensure an outcome that is the best for the whole community.
But in the case of a faculty senate being in favor of pay raises and also being pro-union? Yes, yes I would imagine they are. Whether or not they get a raise is usually not solely determined by a faculty senate, though, or that would be a pretty naive system.