  We consider static and cylindrically symmetric interior string type solutions
in the scalar-tensor representation of the hybrid metric-Palatini modified
theory of gravity. As a first step in our study, we obtain the gravitational
field equations and further simplify the analysis by imposing Lorentz
invariance along the $t$ and $z$ axes, which reduces the number of unknown
metric tensor components to a single function $W^2(r)$. In this case, the
general solution of the field equations can be obtained, for an arbitrary form
of the scalar field potential, in an exact closed parametric form, with the
scalar field $\phi$ taken as a parameter. We consider in detail several exact
solutions of the field equations, corresponding to a null and constant
potential, and to a power-law potential of the form $V(\phi)=V_0\phi ^{3/4}$,
in which the behaviors of the scalar field, of the metric tensor components and
of the string tension can be described in a simple mathematical form. We also
investigate the string models with exponential and Higgs type scalar field
potentials by using numerical methods. In this way we obtain a large class of
novel stable string-like solutions in the context of hybrid metric-Palatini
gravity, in which the basic parameters, such as the scalar field, metric tensor
components, and string tension, depend essentially on the initial values of the
scalar field, and of its derivative, on the $r=0$ circular axis.
