The aim of the paper is to classify the real quadratic number fields $Q(\sqrt{d})$ having specific form of continued fraction expansions of algebraic integer $w_d$ and is to determine the general explicit parametric representation of the fundamental unit $\varepsilon_d$ for such real quadratic number fields where $d \equiv 2,3\pmod{4}$ is a square free positive integer. Also, Yokoi's $d$-invariants $n_d$ and $m_d$ will be calculated in the relation to continued fraction expansion of $w_d$ for such real quadratic fields.