If you are asking specifically about Quine's position, he sticks rigidly to classical logic. He does not accept logical pluralism and does not allow that any contradictions are true. In Two Dogmas (1951), Quine says that logic itself may be revisable in the light of experience, but by the time he wrote Philosophy of Logic (1970), he had retreated away from this position and held that classical logic is the only logic and that anyone who uses anything else is changing the subject.
On his earlier view, a change in logic might be motivated by the usual pragmatic considerations that lead us to favour one theory over another: simplicity, expressive power, consilience, parsimony, lack of adhocness, explanatory power, etc. Maybe a different logic provides a better way of enabling us to describe the world, or at least to describe some particular domain of application. On this view, logic consists of the most tightly held threads in our web of belief, and so are highly resistant to revision, but not immune.
On his later view, only classical logic, specifically first order classical logic, qualifies to be called logic. Quine doesn't believe in 'meanings' but he comes extremely close to saying that classical logic is guaranteed by what logical terms mean, and anyone who uses a different logic is changing the meaning. For myself, I find his earlier position more consistent. Claiming that some sentences, even logical truths, must be true because of their meanings is what Quine criticised in his earlier work, including Truth by Convention (1936), Two Dogmas of Empiricism (1951), Carnap and Logical Truth (1954) and Word and Object (1960).
Moving beyond Quine, holism is not incompatible with logical pluralism. It is common to hold that theories are underdetermined by data. So we may have several rival theories that are consistent with known data, are self-consistent, but are not compatible with each other. A logical pluralist might say the same for logic. Maybe logics are underdetermined by the totality of our experience of the world, and so there are several logics, with no definite criteria by which to say that one is uniquely correct.
A weaker form of pluralism relativises logic to a domain of application, or a case. We may want a logic of truth, a logic of obligation, a logic of necessity, a logic of assertability, a logic of proof, etc., and these turn out to be different logics. On this weaker view, there are lots of territories and each territory needs a different map. On the stronger version of pluralism, there are several maps of a single territory with no way to choose between them.
As to contradictions, most logics consider all contradictions to be false, but dialetheism allows that some are true. Dialetheism is consistent with holism. On the weaker version, there may be domains or cases where we wish to deploy a paraconsistent logic. For example, when studying inconsistent structures in mathematics. And there may be other domains where an explosive logic is appropriate.
On the stronger version of pluralism, things are slightly more difficult, but we may still allow that there are different logics that are capable of accounting for our experience, and some of these are paraconsistent and some not. We do not need to rely on contradictions to revise our assessment of a logic, because we can use the same pragmatic criteria that we use to assess theories.
It is also worth noting that holism does not have to be entirely 'wholly'. Quine himself, in a retrospective, wrote this:
Looking back on it, one thing I regret is my needlessly strong
statement of holism.
The unit of empirical significance is the whole of science [...]. Any
statement can be held true come what may, if we make drastic enough
adjustments [...]. Conversely, [...] no statement is immune to
This is true enough in a legalistic sort of way, but it diverts
attention from what is more to the point: the varying degrees of
proximity to observation, the example of the brick houses in Elm
Street. In later writings I have invoked not the whole of science but
chunks of it, clusters of sentences just inclusive enough to have
critical semantic mass. By this I mean a cluster sufficient to imply
an observable effect of an observable experimental condition.
W. V. Quine. "Two Dogmas in Retrospect". Canadian Journal of Philosophy, Vol. 21, No. 3 (1991), p. 268.