Does a time signature imply accents by default? For example, in 4/4 time should I accent every quarter beat? Should I accent the first beat of each bar?
If yes, how should I notate a piece where these beats should be without accents?
Yes, indeed. The time signature together with an understanding of the musical style of the period in which the music was written tell you what the rhythm is and what the accents are supposed to be in general. Specific notes in specific measures might be written in such a way as to over-ride the default pattern.
For example, in classical music in general, in 4/4 time, the downbeat of the measure, the "1", is the strongest beat, and the other three beats are weaker. However, in rock music in 4/4 time, the strong beats are understood to be "2" and "4".
6/8 is a compound time. There are two beats in the measure, and each beat is further divided into three sub-beats. One beat is represented by a dotted quarter note (three eighth notes) so there are two of those beats in the measure. Among the six sub-beats, the strong beats are "1" and "4", and we generally count them out loud by saying "ONE two three TWO two three".
At the start of every measure, no matter the time signature beat one typically gets a stronger accent than the rest of the notes and any musician either consciously or subconsciously will do this. Every beat will have some kind of accent based on the time signature and this can be extended in certain time signatures like 4/4 and 6/8 where there is a second emphasised beat on beat 3 in 4/4 and beat 4 on 6/8. These are ideas that just come along with the idea of a time signature.
There's really no way around this if you are writing with the structure that a time signature provides as this is the way it is. It's like trying to write exact pitches when the pitch doesn't matter. There are pieces with no time signature like Messiaen's Quartet for the End of Time 3rd section as you can see:
Even still in this the bar line which splits measures is used to show if and when something gets an accent.
A time signature implies a structure of the music, just like spaces in a written sentence imply a structuring into words. When reading a sentence, you don't make an explicit pause after each word, yet the word structure is related to the conveyed meaning, and somebody reading a written text does it subtly different from somebody reading a continuous phonetic transcription.
It's more important that you feel the meter than that you execute it. When accents are written in music, more often than not they indicate some temporarily different/shifted meter rather than single louder notes. When that is the case, it is again more important that the musician feels the changed meter rather than that he "executes" the accents.
So while 4/4 has generally a stronger weight on 1 and a weaker weight on 3 and that's basically the most definite you can say about it generally, that's not all there is to it.
This becomes particularly important with some particular dances where the rhythms are usually rendered unevenly, like a Viennese Waltz where the second and third beat drag somewhat, or swinged rhythms where the off-beat notes are somewhat delayed: in that case the "rules" don't just suggest accents but rather affect the whole rhythmic framework in more or less subtle manners.
No. Accents can be on each on the crotchet beats like you mention. It can be on the weak part of the pulse (Syncopation) There can be no accents. The accent can be on certain quavers. None of which has all that much to do with the time signature.
There is no concrete place where a accent must be. There are places where an accent would be natural and places where it would not be but even when it is not a natural place for an accent it can still be musically valid depending on what effect the music is trying to achieves
There is an implied accent structure in the time signature. Syncopation - accents that DON'T follow this structure - is frequently used. But without the implied underlying structure there can be no syncopation, you can only be "off the beat" when there IS a beat!
If you write using barlines, it can be hard to prevent players from noticing them! There's a direction "Senza misura" meaning "In free time", but that's more about a flexible beat than a lack of accents I think. You can write without barlines completely of course, as in plainchant.
When I first started drums it was because I was sent to a teacher for drums. That was over 50 yrs ago, and as I understand it from my memory a time signature denotes how many notes to a measure and each time signature had accents on different beats in the measure, otherwise if all time signatures had a accent on the 1 and 3 then a 4/4 would sound just like a 2/2. Also I noticed that many don't explain that 3/4 has a quarter note rest where the 4th beat is 1,2,3,-, If I was to try out for an orchestra and didn't know where the accent hits II don't think I would get that job.