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In this paper we investigate Diophantine equations of the form$T^2=G(\overline{X}),\; \overline{X}=(X_{1},\ldots,X_{m})$, where $m=3$ or$m=4$ and $G$ is specific homogenous quintic form. First, we prove that if$F(x,y,z)=x^2+y^2+az^2+bxy+cyz+dxz\in\Z[x,y,z]$ and $(b-2,4a-d^2,d)\neq(0,0,0)$, then the Diophantine equation $t^2=nxyzF(x,y,z)$ has solution inpolynomials $x, y, z, t$ with integer coefficients, without polynomial commonfactor of positive degree. In case $a=d=0, b=2$ we prove that there areinfinitely many primitive integer solutions of the Diophantine equation underconsideration. As an application of our result we prove that for each$n\in\Q\setminus\{0\}$ the Diophantine equation \begin{equation*}T^2=n(X_{1}^5+X_{2}^5+X_{3}^5+X_{4}^5) \end{equation*} has a solution inco-prime (non-homogenous) polynomials in two variables with integercoefficients. We also present a method which sometimes allow us to prove theexistence of primitive integers solutions of more general quintic Diophantineequation of the form $T^2=aX_{1}^5+bX_{2}^5+cX_{3}^5+dX_{4}^5$, where $a, b, c,d\in\Z$. In particular, we prove that for each $m, n\in\Z\setminus\{0\},$ theDiophantine equation \begin{equation*}T^2=m(X_{1}^5-X_{2}^5)+n^2(X_{3}^5-X_{4}^5) \end{equation*} has a solution inpolynomials which are co-prime over $\Z[t]$. Moreover, we show how modificationof the presented method can be used in order to prove that for each$n\in\Q\setminus\{0\}$, the Diophantine equation \begin{equation*}t^2=n(X_{1}^5+X_{2}^5-2X_{3}^5) \end{equation*} has a solution in polynomialswhich are co-prime over $\Z[t]$.

On primitive integer solutions of the Diophantine equation $t^2=G(x,y,z)$ and related results