How can I get EASA Aircraft Maintenance Licence?Is there a training course for getting this licence? I'm an Aircraft Construction and Design Engineer

For EASA AME licence, you must clear the modules first and you must have 5 years of work experience in aircraft maintenance from approved aircraft maintenance organisation (EASA approved experience is preferable) for getting basic maintenance licence.  refer this link https://www.easa.europa.eu/faq/21067

First you will have to find an issuing body that is near you, that issues a Part 66 licence. You will be probably looking at Part-145 or Part-147 bodies. Examinations for the modules do not require training courses if you are already familiar with the subjects. If you are not familiar with all the subjects, you may take full classes, or tutorial classes, but again it's not mandatory, it just increases your chances of being successful. 

After getting your first module, you will have to get all the rest within ten years as each module has a 10-year life span before getting the licence. Lastly, even though you do not need any experience at the time of examination, you will not be given a licence unless you have specific years of experience in the field, which can be proved by your log books.