tl;dr Yes – but read on for more details.
The Schrodinger equation for any system has many solutions, of which only those which are square integrable (aka “normalizable”) correspond to possible states of the system. And to determine a particular solution for any differential equation requires the imposition of “boundary conditions”, which for a Schrodinger equation might take the form of a specification of the function at some particular time (say t=0)
Quantum numbers are just possible values for observables which may be discrete (such as the energy levels for a confined system like an electron in the potential well of a nucleus), or continuous (such as the position and momentum coordinates). And once a particular solution has been identified it does tell us all we can know about what those values will be.
For example, the time-dependent Schrodinger equation for a free particle has “plane wave” solutions, which do not actually correspond to any physical state because they have infinite norm and so to normalize them would reduce them to zero – which means that the particle has zero chance of actually being found anywhere in any finite region. [This is a special case of the uncertainty principle which says that having a precisely defined value of the momentum quantum number implies a completely unknowable position and the other side of this coin would be the “delta function solution” which (is not really a function but) somehow represents the concept of having a precisely defined value of the position and has a momentum-space representation that is completely uncertain with zero probability of being in any finite range of values.]
The evolution of a normalizable “wave packet” solution then does tell how the probability distribution of position quantum number varies over time (and its fourier transform gives the corresponding information about momenta).
For the case of a particle confined by a box or potential well (such as that of an electron in an atom) there may be observables such as the energy (at least when it is low enough) and angular momentum that can only have discrete values. And finding the general solution of the Schrodinger equation will identify these.
So when you solve for the general solution it tells you all the possible values of the quantum numbers but doesn’t say anything about any particular electron. And when you solve for a particular solution (say with some particular initial conditions) then its time evolution will allow you to calculate how the probabilities of having different quantum numbers varies over time.
Furthermore, since energy is conserved, the energy quantum numbers don’t change over time; but once we have calculated all the possible fixed energy solutions, [math]\psi_{n}(x,t)[/math] with energy [math]E_{n}[/math], we can use them to shortcut the solution for a general initial condition by expressing it as a linear combination [math]\Psi(x,0) = \Sigma c_{n}\psi_{n}(x,0)[/math] of the eigenfunctions and multiplying each by the corresponding complex exponential phase factor to get [math]\Psi(x,t) = \Sigma c_{n}\psi_{n}(x,t)=\Sigma c_{n}e^{iE_{n}t}\psi_{n}(x,0)[/math] which allows us to see how the probability distribution for the position quantum numbers evolves over time.