Click on the arrow to the left of the dbo object under Project_3690, and it should show you the tables you have access to. If it does not, then you have a permissions problem and will need to talk with the DBA. That allows you to see them via the GUI. In fact, if you don't already know the names of the tables you should be accessing (such as to follow my code below), then this is the easiest, as they are already filtering out the system and other tables that obscure what you need.
To see them in R code, then dbListTables(conn) will show you all tables, including the ones in the Connections pane I just described but also a lot of system and otherwise-internal tables that you don't need. On my SQL Server instance, it returns over 600 tables, so ... you may not want to do just that, but you can look for specific tables.
For example, if you know you should have tables Table_A and Table_B, then you could do
alltables <- dbListTables(conn)
grep("table_", alltables, value = TRUE, ignore.case = TRUE)
to see all of the table names with that string in its name.
If you do not see tables that you know you need to access, then it is likely that your connection code did not include the specific database, in which case you need something like:
conn <- dbConnect(odbc(), database="Project_3690", uid="...", pwd="...",
server="...", driver = "...")
(Most fields should already be in your connection code, don't use literal ... for your strings.)
One can use a system table to find the other tables:
DBI::dbGetQuery(conn, "select * from information_schema.tables where table_type = 'BASE TABLE' and table_schema = 'dbo'")
# TABLE_CATALOG TABLE_SCHEMA TABLE_NAME TABLE_TYPE
# 1 Project_3690 dbo Table_A BASE TABLE
# 2 Project_3690 dbo Table_B BASE TABLE
# 3 Project_3690 dbo Table_C BASE TABLE
(Notional output but representative of what it should look like.)
