Does a history of conflict between a parent and grandparent affect a contact application's odds?

It can, but not necessarily in a way that dooms the application - courts try to separate conflict between the adults from what's actually good for the child, since the test is the child's best interests, not whose side of an argument was more reasonable. A grandparent-parent conflict rooted in disagreements about parenting choices, lifestyle, or unrelated family disputes generally carries less weight against contact than conflict that reflects a genuine safety concern or a pattern of the grandparent undermining the parent's authority in front of the child.

Where conflict becomes a real problem for a contact application is if it suggests contact itself would expose the child to ongoing tension, poor communication, or a grandparent unwilling to respect the parent's decisions about the child's upbringing - those things can affect whether, and how, contact is structured, sometimes leading to more limited or supervised contact rather than outright denial. If conflict is a significant factor in your situation, focus on showing the court how contact can be arranged in a way that shields the child from the adults' disagreements, rather than trying to relitigate who was right in the underlying conflict.