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We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) methods tostudy the phases and transitions displayed by a class of sign-free designerHamiltonians for SU($N$) analogs of spin $S=1$ quantum antiferromagnets on thesquare lattice. The SU($N$) spins are generators of the single-row two-columnrepresentation (complex conjugate of single-row two-column representation) on$A$ ($B$) sublattices of the square lattice, and the Hamiltonian is designed toexplore the competition between the nearest neighbour antiferromagneticexchange couplings $J$ and four-spin interactions $Q$ that favor aplaquette-ordered valence bond solid (p-VBS) ground state. We find that thisstate is indeed established at large $Q/J$ for all $N > 3$. For $3< N \leq 9$,the ground state exhibits a direct first order quantum phase transition from asmall-$Q/J$ N\'eel ordered antiferromagnetic state to this large-$Q/J$ p-VBSstate. The ground state at $Q/J=0$ for $N \geq 10$ has been previously reportedto be a valence bond nematic state, dubbed the Haldane nematic in recentliterature. For small nonzero $Q/J$ and $N \geq 10$, we additionally find anunusual intermediate state in which the bond energy has a Bragg peak atwavevector $(\pi,\pi)$ with no accompanying Bragg peaks at wavevectors$(\pi,0)$ and $(0, \pi)$. This $(\pi, \pi)$ state also appears to be metastableat $Q/J =0$, as evidenced by low temperature histograms of the Haldane nematicand $(\pi, \pi)$ order parameters. Deep in the p-VBS phase of the ground statephase diagram, we find the temperature-driven melting of the p-VBS order is inthe Ashkin-Teller universality class. In this regime, we identify aninteresting signature of Ashkin-Teller criticality in bond correlations atwavevector $(\pi, \pi)$; this is in addition to the expected criticalfluctuations of the conventional p-VBS order parameter at wavevectors $(\pi,0)$and $(0,\pi)$.

Competition between Neel, Haldane nematic, plaquette valence bond solid,  and $(\pi,\pi)$ valence bond solid phases in SU(N) analogs of $S=1$  square-lattice antiferromagnets