I suspect you want:
var query = from entry in dbContext.Table1
where entry.Department == "ABC" && entry.SkillName != null
select entry.SkillName;
var count = query.Distinct().Count();
Or using extension method syntax, in one go:
var count = dbContext.Table1
.Where(entry => entry.Department == "ABC" &&
entry.SkillName != null)
.Select(entry => entry.SkillName)
.Distinct()
.Count();
As shown by mesiesta, you can combine query expressions with calls not supported within query expressions, but I tend to assign the query expression to an intermediate variable... I personally find it clearer, but use whichever you (and your team) prefer.