I think you mean something different here as the birefringence of objective lenses is nominally zero. Objective lenses are typically made from glasses, which are amorphous, and should therefore exhibit no birefringence at all. If not properly mounted the individual lenses in the objective may certainly exhibit birefringence due to mechanical stress, but this should be carefully avoided. A nice visualization of stress birefringence is here: https://www.sciencephoto.com/media/569574/view/stress-birefringence-in-a-plastic-cup

I assume that you would like to compute how an objective lens alters the beam polarization, not just the objective lens birefringence. In principle, you can do the both. For computation of birefringence distribution you have to know the type of glass of each lens and the mechanical stress distribution. Then using a specific stress optical coefficient for each glass type you can compute the lens birefringence. In order to find how the lens birefringence modifies the polarization you have to compute optical indicatrix field, make ray tracing, etc. But the polarization change is also caused by Fresnel reflection on the lens surfaces, by multilayer antireflection coatings, etc. Usually the objective lens design is confidential and this information is not readily available. Therefore you might to consider an alterative way and to measure the polarization aberrations and the extinction coefficient of your objective lens:

https://www.researchgate.net/publication/260926496_Polarization_aberrations_caused_by_differential_transmission_and_phase_shift_in_high-numerical-aperture_lenses_Theory_measurement_and_rectification

The basic ingredients that are needed to detect low levels of birefringence (retardance < 10 nm) are high-extinction optics, use of low-retardance compensator, light source with high irradiance, and high-sensitivity detector (e.g., dark adaptation for visual observation and measurements). The need for high extinction optics applies to all components of the polarization optical train, which starts and ends with the polarizer and analyzer, respectively, and all optical components placed between them. Most manufacturers carry objective and condenser lenses that are either made or specially selected for polarized light observations. Such objectives typically carry the designation P, PO, or POL on their lens barrel and are designed to induce minimal polarization aberrations.