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Let $S$ be the collection of quadratic polynomial maps, and degree$2$-rational maps whose automorphism groups are isomorphic to $C_2$ definedover the rational field. Assuming standard conjectures of Poonen and Manes onthe period length of a periodic point under the action of a map in $S$, we givea complete description of triples $(f_1,f_2,p)$ such that $p$ is a rationalperiodic point for both $f_i\in S$, $i=1,2$. We also show that no more thanthree quadratic polynomial maps can possess a common periodic point over therational field. In addition, under these hypotheses we show that two nonzerorational numbers $a, b $ are periodic points of the map $\phi_{t_1,t_2}(z)=t_1z + t_2/z$ for infinitely many nonzero rational pairs $(t_1, t_2)$ if and onlyif $a^2 = b^2$.

Simultaneous Rational Periodic Points of Degree-2 Rational Maps