  We prove an effective density theorem with polynomial error rate for orbits
of upper triangular subgroup of $\mathrm{SL}_2(\mathbb{Q}_p)$ in
$\mathrm{SL}_2(\mathbb{Q}_p) \times \mathrm{SL}_2(\mathbb{Q}_p)$ for prime
number $p > 3$. The proof is based on the use of Margulis function, a
restricted projection theorem on $\mathbb{Q}_p^3$, and spectral gap of the
ambient space.
