This is a complicated question that doesn't have an exact answer. In general, yes, the quality will probably be lower but the higher frame rate is higher "quality" to begin with.
With video, you have to remember that temporal information is part of the quality. If you double the frame rate, the quality of each individual frame will go down, but you will see twice as many of them and the impact of things like noise can average out amongst the frames.
Additionally, modern compression works by comparing one frame to neighboring frames. When you increase the frame rate, the amount of change between frames is reduced, so the amount of data needed to store the change is also reduced, since less changes between frames.
Purely theoretically, what most matters in terms of quality is the amount of information being put in front of our eyes over time, so it may even be theoretically possible to make a higher frame rate version of a video that is higher overall quality at the same bitrate as a lower frame rate version.
With a hypothetical perfect encoding and compression system, the rate of meaningful information presented to our hypothetical perfect eyes per second is what determines quality. The problem is that such hypothetical don't exist. Our eyes are unpredictably lossy in terms of what information they pay attention to over time, so in some cases higher frame rate provides a bigger boost in quality than others. Similarly, compression and encoding isn't perfect either, so some types of content require more effort to encode and compress, with more overhead, and others produce more noise that is best accounted for through higher frame rates, while others may require lower frame rates but higher detail per frame.
Practically, the overhead of storing parts of the frame that can't compress easily generally overcome the advantages of the elevated frame rate resulting in a loss of quality when doubling frame rate without increasing bit rate, but it isn't anywhere near as significant of a quality loss as you would get if you cut the bitrate in half.
Additionally, the quality loss of the compressed version over the original is much higher for the higher frame rate, even if the relative quality between the lower and higher frame rate versions are much closer (since the lower frame rate video was much lower quality to begin with).
You only need to increase the bitrate enough to make up for the overhead lost by inefficiencies in the encoding and compression of the extra frame data. As ProfessorSparkles pointed out, somewhere around 1/3 extra bitrate is probably a decent starting point for figuring out what you need to maintain subjective quality, but it will vary a lot based on the video content you are encoding and the codecs and data rates being used.