  A classic result in extremal graph theory, known as Mantel's theorem, states
that every non-bipartite graph of order $n$ with size $m>\lfloor
\frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab.
Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for
given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a
spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral
versions of Mantel's theorem.
