  In this paper we prove a $q$-analogue of Koshy's formula in terms of the
Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in
terms of $q$-hypergeometric series due to Andrews by applying the
inclusion-exclusion principle on Dyck paths and on partitions. We generalize
these two $q$-analogues of Koshy's formula for $q$-Catalan numbers to that for
$q$-Ballot numbers. This work also answers an open question by Lassalle and two
questions raised by Andrews in 2010. We conjecture that if $n$ is odd, then for
$m\ge n\ge 1$, the polynomial $(1+q^n){m\brack n-1}_q$ is unimodal. If $n$ is
even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial
$(1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the
second problem posed by Andrews.
