Usually in formal writing, it's safest to actually write out the word you're using.
Any of the following are correct.
"a quarter of the people voted for..."
"one fourth of the people voted for..."
"a fourth of the people voted for..."
In your other example though "a 1/4 inch hole" could be correctly written as "a one-quarter inch hole". In this case "one-quarter" is an adjective describing a hole where the article belongs to your noun. However, if it's completely unavoidable you should presume people will read it as "one quarter" or "one fourth" rather than "quarter" or "fourth" and leave off the article.