Job Application Help?

I am not satisfied with my current job and really hope to look for another job. I just graduated from my university. I wanted to stay in this job while searching for a suitable one. In your opinion , will the company that I am applying to regard this as a sort of betrayal or contact my current employer to find out about me? If that is the case, I am afraid that my current company will sack me . I need the income to support myself that why I need to stay while in the look. Any suggestion on this ? Thank you.