Yes, it is possible to perform an SQL injection attack without supplying quotes in the parameter.
The way to do this is with an exploit to do with how numbers and/or dates are processed. You can specify at the session level what the format of a date or number is. By manipulating this you can then inject with any character.
By default in the UK and US, a comma is used to indicate the thousands separator in numbers, and a full stop for the decimal point. You can change these defaults by executing:
alter session set nls_numeric_characters = 'PZ';
This means that "P" is now the decimal point and "Z" is the thousands separator. So:
0P01
Is the number 0.01. However, if you create a function P01, the object reference will be picked up before number conversion. This allows you to execute functions on the database giving you increasing powers, as follows:
Create a basic "get by id" function:
create procedure get_obj ( i in number ) as
begin
execute immediate 'select object_name from all_objects where object_id = ' || i;
end;
/
Also create a function P01 which does something undesirable (in this case just creating a table, but you get the idea):
create function p01 return number as
pragma autonomous_transaction;
begin
execute immediate 'create table t (x integer)';
return 1;
end;
/
And we're good to go:
alter session set nls_numeric_characters = 'PZ';
SELECT * FROM t;
SQL Error: ORA-00942: table or view does not exist
exec get_obj(p01);
anonymous block completed
SELECT * FROM t;
no rows selected
No quotes anywhere, but we've still managed to execute the "hidden" function P01 and create the table t!
While this may be difficult to do in practice (and may require some internal knowledge/help), this does show that you can inject SQL without having to have quotes. Altering the nls_date_format can allow similar things to be done.
The original findings for numbers were by David Litchfield and you can read his paper here. You can find Tom Kyte's discussion of how dates can be exploited here.
Best Answer
Since you log all queries against the database, look for queries that don't belong.
Start with queries that failed. Assuming your app isn't riddle with bugs, there should be very few queries submitted to the DB that fail. Attackers typically have a set of queries they execute once they find a vulnerable app. If you apply the principle of least privileges on accounts used to access the DB, you should find a bunch of failed queries. There will also be queries that fail because a less sophisticated attacker might have a bunch of queries that are not compatible with your database edition, version or vendor.
Next, look for queries/commands that could not have been generated by your application. For instance, if the application should never accesses system tables, system functions, information schema, etc... but you see them in your log, that's a good indicator something might be amiss and warrants deeper investigation. Looking at your OS logs might be helpful here in case they tried or succeeded in planting a backdoor/rootkit.
Once you're done with that, look for queries with unusual/unexpected behavior. For instance, queries that return significantly more data than normal. If you see queries that access all tables around the same time or ran show tables then accessed each table in order, there's a good chance you have an attacker that tried (or succeeded) to exfiltrate data. Even something seemingly innocuous like SELECT * without a WHERE clause is suspicious (assuming your app doesn't do this). Note that this applies to any possible query, not just SELECT. The attacker might be some disgruntled employee and decided to randomly DELETE or UPDATE data to get you in trouble with your auditors or worse.
Looking at IP addresses is often not helpful even if you're looking at the web server logs unless you have a narrow and/or well known band of IP addresses your users connect from. Few attackers do it from their home PC anymore. Bots and proxies are found all over the world plus IPs are easily spoofed. Unless you get your ISP or Telco provider involved, it's unlikely you'll get very far looking at your own logs. One better indicator is your corporate firewall or network logs to find unusual outbound IP targets. An attacker that exfiltrates data has to send it to some real endpoint. That is usually to a place that isn't cooperative and/or moves slowly when foreign companies or law enforcement comes knocking.