Question: Does Java support QoS (Quality of Service) for packets that are sent over the Internet?

Answer: Yes, Java supports although it also depends on network configuration...
Paths through the Internet vary widely in the quality of service they provide. Some paths are more reliable than others. Some impose high call setup or per-packet charges, while others do not do usage-based charging. Throughput and delay also vary widely. Often there are tradeoffs: the path that provides the highest throughput may well not be the one that provides the lowest delay or the lowest monetary cost. Therefore, the "optimal" path for a packet to follow through the Internet may depend on the needs of the application and its user.

Because the Internet itself has no direct knowledge of how to optimize the path for a particular application or user, the IP protocol provides a (rather limited) facility for upper layer protocols to convey hints to the Internet Layer about how the tradeoffs should be made for the particular packet. This facility is the "Type of Service" facility, abbreviated as the "TOS facility" here.
Although the TOS facility has been a part of the IP specification since the beginning, it has been little used in the past. However, the Internet host specification now mandates that hosts use the TOS facility. Additionally, routing protocols (including OSPF and Integrated IS-IS) have been developed which can compute routes separately for each type of service. These new routing protocols make it practical for routers to consider the requested type of service when making routing decisions.

So, in Java you can try use the method setTrafficClass(int tc) of Socket class. As the underlying network implementation may ignore this value applications should consider it a hint.
For Internet Protocol v4 the value consists of an octet with precedence and TOS fields as detailed in RFC 1349. The TOS field is bitset created by bitwise-or'ing values such the following:-

IPTOS_LOWCOST (0x02)
IPTOS_RELIABILITY (0x04)
IPTOS_THROUGHPUT (0x08)
IPTOS_LOWDELAY (0x10)
read more in RFC 1349 and Java API

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