We know that in company $Y$ of 37 employees, the number of employees having birthdays in July is more than the number of employees having birthdays in any other month; we need to find the number of employees of the company Y who have birthdays in July. so that at least 3 employees have their birthdays in each month.

If we have 3-3 employees for each of the 12 months, the number of employees who will left will be $\(37-(3 \times 12)=37-36=1\)$, so to have more number of people having birthday in July when compared with other months, this one remaining person must have birthday in July.

So, the number of people having birthday in July has to be $\(3+1=4\)$.
Hence the answer is (A).