public with sharing class MyCLass {

public Contact fakeCont { get; set; }
public String queryString { get; set; }
public MyCLass() {
    fakeCont = new Contact();
}

public void doQuery(){
    queryString = 'SELECT id, Name FROM SObject WHERE dateExample = ' + fakeCont.Birthdate  ;
}

<apex:inputField styleClass="form-control" value="{!fakeCont.Birthdate}"/>