Sure.
The Martini is a drink. (you can read the history on wiki)
It is made with gin and vermouth. Only. Ever. (in differing proportions depending on where you are, in history)
A cocktail is a category of drink. It is usually distinguished by having at least two liquid ingredients, one of which is alcohol. Although two is more likely. Ingredients here can also include sweeteners, juices, aromatized wines, bitters, etc.
On a seperate topic, and one that, I believe, is more in line with the question here is the “Cocktail Glass”. To begin, and I cannot stress this enough, there is no such animal as a “Martini Glass”. Period. The piece of stemmed barware characterized by a conic bowl is called a “Cocktail Glass”. (not to be confused with the category of drink mentioned earlier because cocktails can come in various glasses.)
Unless I’m wrong, what the poster wants is a fruity cocktail served chilled, up, in a cocktail glass. I’d second Dave’s suggestion of The Maiden’s Prayer because it is flavorful, somewhat sweet, and also somewhat challenging to the palate. A cocktail shouldn’t be able to be guzzled like fruit punch. It should be sipped, savored.
That being said, If you want chuggable buy some inexpensive raspberry flavoued vodka and shake with a little cranberry and oj and strain into a chilled cocktail glass and get some drink on. :)