There do not appear to be any quotations by anyone specifically saying 'giving women the vote would be like giving cows the vote' although that does not mean that no one did. These words (or very similar) have been used together on a number of occasions, though.
As it is not uncommon for people / sources to be misquoted or misinterpreted (see 6 Famous Literary Quotes Everyone Uses Exactly Wrong and List of Misquotations), misunderstanding may be the root of this ‘quote’ comparing women and cows in voting.
In addition to the Frederick Douglas quote cited by Wladimir Palanthe, the idea of someone saying something along the lines of ‘giving votes to women would be like giving votes to cows’ may possibly stem from one of the following:
By raining themselves higher, as “woman in herself,” as the “higher woman,” as a female “idealist,” they want to lower the level of the general rank of women; and there is no surer means for that than higher education, slacks, and political voting – cattle rights.
Higgins notes that this has been taken to mean (wrongly, in her view) that “women’s votes are discounted by the reference to cattle” by the writer and academic Carol Diethe, so it is not inconcievable that others misinterpreted Nietzsche as comparing women to cows on the topic of voting.
Another possible misrepresentation could stem from this postcard using an adapted Edward Lear limerick published by Suffrage Atelier in 1913, a publication which campaigned for women’s suffrage. The postcard “refers to parliamentary procrastination in relation to the introduction of a female suffrage bill.” but without reading the text very carefully, an entirely different meaning is quite possible (and the anti-suffrage movement also produced many postcards).