Not necessarily. In full sunlight you will not be able to tell any difference in quality. You will notice the crop camera will not give you as shallow a depth of field in focus as a full frame camera would. Sometimes you want a shallower depth of field, but then again sometimes you don't.
The image quality difference would appear most dramatically when shooting in dimmer light, if you need to raise the ISO to get your shutter speed up. The crop camera will show noise, grain and loss of detail a stop or two worse than the full frame body will.
One of the underlying truths is that the similarities are greater than the differences. The other day while I was steering my cousin's sailboat, my wife took about a dozen pictures with my 5D3, thinking it was her 7D2. (She was a bit confused by my different settings and changed some of them and the battery to try to straighten it out, but that's just an implementation detail. The pictures we took with both cameras are indistinguishable in quality, given the almost ideal shooting conditions.)
Do full frame cameras produce better quality images than a crop sensor camera?
That would depend upon your definition of quality. Under your most typical shooting conditions, there is little difference in image quality, which is yet another subjective metric. Both sensor types can render high quality images under a variety of conditions, too.
One difference is depth of field. Provide a full frame camera body and an APS-C camera body lenses of equal focal length, the full frame body will produce narrower depths of field. But, when you use lenses that provide an equivalent angle of view [50mm on an APS-C sensor, and 80mm on a FF sensor] with each type of sensor, then the APS-C sensor will have the narrower depth of field.
The most apparent differences between the quality of the images reproduced by each type of sensor occurs when you push the camera to its' performance limits. This is where your shooting styles, habits, and subject matter factor into play. Many still photographers prefer full frame sensors because they typically have larger sensing element that can collect more light, which translates to better contrast and color. Many action photographers prefer APS-C sensors because of the "extra reach" that the smaller sensor can give them.
Which type of sensor is better for you really depends upon your personal preferences and needs.
For some folks, like the recent post about getting a person and airplane both in focus at the same time want *large* depth of field.
"Do full frame cameras produce better quality images than a crop sensor camera?"
You can believe, or not, all the BS going around endlessly on the inner web about crop vs FF but the simple truth is, it is all just numbers. None of what you just read above is true if you put the correct numbers to it.
The most important number is pixel size. It is not sensor size as most folks would lead you to believe. It is also important to remember what cameras models you are comparing. This fact is hardly ever considered.
A better way to understand this is to consider a lens' "angle of view" (AOV), sometime referred to as "angle of acceptance". If this number is matched the resulting photo will be the same. No matter what camera you are using. That includes crop, FF or even medium format cameras. Now add the pixel size factor. If the pixel size is the same or nearly the same it will be virtually impossible to tell a FF from a crop sensor.
You may draw some conclusions from this. If you want high resolution you want to find a camera with a sensor with small pixels. If on the other hand you want low light performance you might look for a sensor with larger pixels. Now reality, It is true that manufactures usually put smaller pixels on a crop sensor to get better resolution but because of less physical room to place them, they are smaller. Likewise, FF cameras have more room available for as many pixels as the cropper but they can be larger. That is the difference!
Lets carry this further, a Full Frame camera with 36 MP will have similar sized pixels to a cropper with just 16 MP. Now add to this realization that there will be far more pixels on the FF sensor and you will now see a big difference. The larger sensor is going to capture more total light. More total light is easier for the downstream electronics to deal with. Does that make it better? Maybe!
I personally own and shoot both a FF and a pseudo cropper a 1.3 (APS-H) sensor. I use the one that's numbers work to my advantage.
So remember, it is just all about the numbers.