We examine the structure of the periodic continued fractions of square roots of non-square positive integers given by an integer-valued quadratic polynomial Q(n) = (an + b)² + (gn + h). The aim is to identify repeated blocks of partial quotients in the period. The quotients in the period form a palindrome, and when the period length is even, the period has a central term a_n. The paper focuses on periods with a_n = a₀ or a_n = a₀ – 1, where a₀ is the initial partial quotient. For a_n = a₀ we give an algorithm to obtain formulas involving repeated blocks comprising three or more elements, not all equal.