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Hi and thanks for the query.
No, the allergy and the coughing is definitely not due to the presence of the implants and is most likely due to an irritant or allergen like dust etc.
The vibration of the upper right side should be checked by your dentist. Implants in the maxilla (upper jaw) are under the influence of gravity all the time and over time, bone resorption can cause the sockets in which the implants are screwed to loosen and cause vibration whole walking or jerky movements.
Your dentist can evaluate and give a definite answer in just one visit and an x ray can confirm, if required.
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Wish you good health.