It is a conjecture of Stieltjes, apparently still open, see

T.J. Stieltjes, Letter No. 275 of Oct. 2, 1890, in Correspondance d'Hermite et de
Stieltjes, vol 2, Gauthier-Villars, Paris, 1905.

that Legendre polynomials of different degrees have no common roots, except $x=0$ when both degrees are odd.

Laguerre polynomials $L_{n}^{\alpha}$, $\alpha>-1$, orthogonal with respect to the weight function $x^{\alpha}e^{-x}$ on $(0,\infty)$, may have common roots. For instance, as observed in

K. Driver, K. Jordaan, Stieltjes interlacing of zeros of Laguerre polynomials from different sequences. Indag. Math. (N.S.) 21 (2011), 204-211.

$L_{2}^{23}(x)$ and $L_{4}^{23}(x)$ have $x=30$ as a common root.
It seems also to be an open question how many common
zeros are possible in general for $L_{n}^{\alpha}$ and $L_{n+m}^{\alpha}$, $n\geq 0$ and $m>1$. The corresponding question for Hermite polynomials about a common root other than zero seems to be unanswered as well.

Finally, a classical theorem of Stieltjes states that if $p_{0},p_{1},p_{2},\ldots$ is any sequence of orthogonal polynomials then the zeros of $p_{k}$ and $p_{n}$, $k<n$, are interlacing in the sense that each open interval of the form
$$(-\infty, z_{1}), (z_{1}, z_{2}), . . . , (z_{k-1}, z_{k}), (z_{k} , \infty)$$
where $z_{1}<z_{2}<\cdots<z_{k}$ are the zeros of $p_{k}$, contains at least one zero of $p_{n}$. It implies that at least $k+1$ zeros of
$p_{n}$ are distinct from the zeros of $p_{k}$, or equivalently $p_{k}$ and $p_{n}$ have at most $\min(k,n-k-1)$ zeros in common. It is proved in

P.C. Gibson, Common zeros of two polynomials in an orthogonal sequence, J. Approx. Theory 105 (2000) 129-132.

that this bound is sharp in the sense that for any $k<n$, there exists a sequence of orthogonal polynomials $q_{0},\ldots,q_{n}$ such that $q_{k}$ and $q_{n}$ have precisely $\min(k,n-k-1)$ zeros in common.