  The three-body Schr\"odinger equation of the H$_2^+$ hydrogen molecular ion
with Coulomb potentials is solved in perimetric coordinates using the
Lagrange-mesh method. The Lagrange-mesh method is an approximate variational
calculation with variational accuracy and the simplicity of a calculation on a
mesh. Energies and wave functions of up to four of the lowest vibrational bound
or quasibound states for total orbital momenta from 0 to 40 are calculated. The
obtained energies have an accuracy varying from about 13 digits for the lowest
vibrational state to at least 9 digits for the third vibrational excited state.
With the corresponding wave functions, a simple calculation using the
associated Gauss quadrature provides accurate quadrupole transition
probabilities per time unit between those states over the whole rotational
bands. Extensive results are presented with six significant figures.
