Since I need to count the number of publications published in a particular topic , how can I find the exact number published in Research Gate at particular time period
Unfortunately, it is not possible to search for topics. RG's search function (https://www.researchgate.net/search.Search.html?query=&type=publication) is very elementary. I guess it searches only for words in the titles and the abstracts.
Although the search function is rudimentary, as suggested by Dr Wolfgang R. Dick , I would still attempt to use it (from the homepage) and click on "research" tab prn.
I would also use the Pubmed search engine as I have generally found it more reliable.
If you need the number of articles published on RG, I don't know. I once had to estimate the number of articles published in a given year on the subject of "trust". For this I used EBSCO and after several adjustments I managed to give an approximately good estimate.