Is there a way to determine the production method of a Higgs boson created via collisions in the LHC? So I'm aware that there are several currently accepted production methods for Higgs boson, such as gluon fusion. When a Higgs boson is detected is it possible to determine the production method from either the detection of the Higgs or from measured information about the decay products/incoming particles. If so do we have a ratio of production methods?
 A: The experiments in the high energy collisions are in the realm of quantum mechanics, and the models to fit the data are quantum mechanical. In quantum mechanics only probability distributions can be  mathematically predicted and checked against the data. On a single event basis, conservation laws apply, energy, momentum angular momentum and all the quantum number conservations, but there is no prediction on the values of the four momenta of single events.

When a Higgs boson is detected is it possible to determine the production method from either the detection of the Higgs or from measured information about the decay products/incoming particles.

Not for a single event.  An event in the Higgs mass peak may be the result of a  number of possible interactions  but it is only the summed cross section that is calculable. The distributions are fitted to the sum of the possible theoretical interactions, including background. Models are checked by fitting the plots,  an elaborate work as seen for example here.
