Yes. This is a standard construction in British English, but not in American English. If you're using 'holiday' as an uncountable noun meaning 'annual leave entitlement from work', this phrase is fine.
But note that this is because of a regional difference in vocabulary, not grammar. The word 'holiday' has a wider range of uses in British English than it does in American English. To prove that there is nothing wrong with the basic underlying grammar of your phrase, here are some similar constructions:
A month's holiday
A week's wages
A day's work
Note that the singular indefinite article above refers to the singular period of time (a month, a day, a week). This is why we say "a week's wages", even though 'wages' is plural. It is also why "I'm taking two months' holiday" has no article. Likewise, "I lost a day's work" but "I lost three days' work": the article agrees with the time period.
As Rady suggests, you could also say 'a one-month holiday'. This is a slightly different meaning of the word 'holiday', as used in AmE. Here, 'holiday' is a countable noun, referring to a specific break from work. For example, "Christmas is a one-week holiday in many parts of the world". When used in this sense, 'one-month' is an adjective, so the article refers to the countable form of the noun 'holiday'. This is why we say 'a one-month holiday' but also 'a two-month holiday': we're talking about one holiday, so we use a singular article.
Note that if this refers to time away from home (a trip abroad with your family, say), this would be 'a one-month holiday' in British English, but a 'one-month vacation' in American English.
I hope that makes sense. I know that Russian-speakers struggle with articles, and this particular example is actually quite complex. When you're dealing with two varieties of English and a word which has several different meanings (both countable and uncountable), it's bound to get complicated.