The team announced that Quessenberry experienced fatigue and a persistent cough over the past week and was diagnosed after seeking a medical evaluation to find out the cause.
"Im gonna kick the crap outta this thing and come back even stronger! I will win this fight!" Quessenberry told CSN Houston.
"He's a guy that wanted his teammates to know first that he was going to fight this thing and he was going to beat it," Texans coach Bill O'Brien said told reporters, per ESPN.com. "The one thing that's stood out to me and our coaching staff ... is what an impressive young man this is."
O'Brien added: "We want everybody to know, especially David Quessenberry, that we're very much in David Quessenbery's corner."
A sixth-round pick in 2013, Quessenberry missed all of last season with a broken foot.
UPDATE: Quessenberry addressed his illness Tuesday on Twitter:
I have been diagnosed with cancer. My fight has just begun. As long as I have breath in my lungs I will fight this disease, and I will win.â David Quessenberry (@DavidQberry) June 10, 2014