  Given a quadratic polynomial with rational coefficients, we investigate the
existence of consecutive squares in the orbit of a rational point under the
iteration of the polynomial. We display three different constructions of
$1$-parameter quadratic polynomials with orbits containing three consecutive
squares. In addition, we show that there exists at least one polynomial of the
form $x^2+c$ with a rational point whose orbit under this map contains four
consecutive squares. This can be viewed as a dynamical analogue of a question
of Fermat on rational squares in arithmetic progression. Finally, assuming a
standard conjecture on exact periods of periodic points of quadratic
polynomials over the rational field, we give necessary and sufficient
conditions under which the orbit of a periodic point contains only rational
squares.
