I see two main questions: how much of the traffic can be attributed to immigrants (although the question also mentions "migrant"; the former generally means people from other countries, while the latter could include Australians who are not native to Brisbane), and how should "cause" of traffic jams be allocated on that basis?
For the first point, the only data that is readily available is the percentage of immigrants. According to wikipedia, "The 2016 census showed that 32.2% of Brisbane’s inhabitants were born overseas." So barring a significant variation between immigrants and natives in driving patterns, the default conclusion is that immigrants are the source of about 1/3 of the traffic. And unless such differences are massive, immigrants are clearly not the sole source of traffic (and if they were the only people driving on the roads, then they would be the only ones suffering from traffic jams anyway). So at this point, there is a very large burden of proof if anyone wishes to claim that immigrants are a majority source of traffic.
Considering the second point: Suppose you have a city with 700k inhabitants with enough roads for 800k inhabitants. Now 200k immigrants move in. Since there are 100k more people than road capacity, there are traffic jams. Did the immigrants cause the traffic jams? Without them, roads would be under capacity. If by "cause", you mean "Without them, this wouldn't happen", then they caused the traffic jams. But if you mean "If it were just them, this would happen", then they didn't. And why single them out? How are the 700k original inhabitants any less a cause of the traffic jams?
Regarding criticisms of my answer:
"The central argument of this answer is theoretical in nature." For the first issue, I have presented well-sourced data that while not being entirely conclusive, is material to the question. The second issue is not a factual matter to begin with, and I am not attempting to give an answer to it so much as explain how it is not a factual issue.
"[You are assuming] That immigrants drive as much non-immigrants." No, I am presenting facts from which the default conclusion is that immigrants are the source of 1/3 of the traffic. Saying "you don't know whether there are other factors" is an argument from ignorance.
"[You are assuming] That immigrants don't contribute as much to road-building as their road-use". If there are insufficient roads for the traffic, then overall road use is, in some sense, greater than road building. So if immigrants are comparable to natives, then both groups must be using roads more than contributing to building them. As to whether they are indeed comparable, that is just a repeat of point 2.
"[You are assuming] That the roads were undercapacity before considering immigrants." If roads were under capacity without immigrants, then there would be traffic jams without them, which would make it even more difficult to assign "blame" to them. Accusing someone of focusing on conditions least favorable to their position is quite an odd charge to make.
"[You are assuming] That immigrants were not considered during road-planning." This is quite nonsensical. None of my points depend on this.
"[You are assuming] That the population of Brisbane has grown only due to immigration." My discussion of the first issue does not deal with population growth at all. With regard to the second issue, having population growth due to sources other than immigration would only weaken the case for immigrants being "to blame", so this is once again criticizing me for focusing on the strongest case against my position.
"[You are assuming] That there isn't any other cause to congestion than simply population rise." The question of whether there are factors that differ between immigrants and natives is simply a repeat of point 2. Any factor that interacts with natives the same as with immigrants does not affect any of my points.