I did just that with a lot of persistence and a bit of luck (of the good variety) thrown into the mix. I was initially hired to do basic hardware upgrades and help out with some user and network support. I knew nothing about networks at the time but quickly learned the environment. Cleaning up after infections led me to looking into security and it grabbed my attention. Every evaluation I had after that I asked for security training and the possibility of a security position. I had strong arguments for both. It took a long time (years) but a restructure exposed me to upper management. I volunteered for projects and learned how to do some technical writing. A 40 page instruction manual wowed a number of VIPs.
In the year that I refer to as "The year of rotating supervisors" someone finally listened to me and I was given the chance to speak with someone in security. The phone conversation was less than a few minutes. I admitted my weaknesses up front and was hit with a single question: "Why security?" I told this person that I was going into the field and I could do it for them or for someone else but that was the path I had chosen because I wanted to do it. A gamble for sure and it came up aces for me.
Now my career is finally going where I wanted it for so long. I do not have certs (yet) or anything beyond a two year degree in business. However, I let people know where I wanted to go and that I was going to get there.
Take the opportunities as they come up and let your supervisor know that you want to move into a security related field.