Indifference curves derived from a quasi-linear utility function are convex if they exhibit a diminishing Marginal Rate of Substitution (MRS), which means they become flatter as one moves to the right along the curve. Since the MRS for such preferences is given by the formula $v'(t)$, the condition for convexity is that $v'(t)$ must be a decreasing function of $t$. In other words, as the quantity of good $t$ increases, the value of the first derivative of its utility component, $v'(t)$, must fall.