For the natural two parameter filtration $(\mathcal{F}_λ: λ\in P)$ on the boundary of a triangle building we define a maximal function and a square function and show their boundedness on $L^p(Ω_0)$ for $p \in (1, \infty)$. At the end we consider $L^p(Ω_0)$ boundedness of martingale transforms. If the building is of $\text{GL}(3, \mathbb{Q}_p)$ then $Ω_0$ can be identified with $p$-adic Heisenberg group.