If you are treating the bird with Doxycycline because you know the bird is sick, it is utterly pointless giving it garlic as well!
If the bird is not sick, it is utterly pointless giving it Doxycycline!
Edit: I think you do need to go to one authorative source, such as Dr Colin Walker, and do some research. When it comes to things like garlic and ACV you will get all shades of opinion on here, or on other bulletin boards. It is probable that what can be said about them is that they may help the bird's immune system in regard to some potential probems. Those are the operative words: may help. There's no guarantee that they will have any dramatic effect, they won't magically give a bird immunity against everything and anything (like viruses, for example), and they certainly won't cure an illness a bird already has!
While allicidin (garlic) is an antiviral, antifungal and natural antibiotic, it should be first use as a preventative, and medication used to treat a specific diagnosed disease. Also, your common kitchen variety garlic is not the same used in some human grade products, it is different and does NOT have healing properties. If you use any herbal compound or homeopathic, please seek a qualified avian vet who is trained in both holistic and traditional.