If I encrypt some data with a randomly generated Key and Initialization Vector, then store all three pieces of information in the same table row; is it necessary to encrypt the IV as well as the Key?
No, only the key may need to be stored encrypted; it is not necessary to encrypt a random IV.
Obviously the (secret) key needs to be kept secret. If you do that using encryption (i.e. shifting the burden to another key or key pair) or any other method is up to the architect. However, if you are going to store it next to the database then a good key wrapping method such as AES-WRAP, AES-SIV is a logical choice as part of securing your key at rest. Other security mechanisms such as access control / protection against side channel attacks need to be in place when using the key.
The IV needs to be unpredictable to an adversary for CBC mode. If that's not the case then CBC fails under an active attack using a ciphertext oracle. In general you should assume that such an oracle exists. For instance, you should assume that the attacker can send any messages to your DB, which you then encrypt and store in the database for the attacker to read back. Having a random IV generated by a well seeded cryptographically secure (pseudo-) random number generator (CSPRNG) - such as /dev/urandom - avoids this kind of problem.
Encrypting the IV is dangerous practice as the IV gets XOR'ed with the plaintext. If you do so you should encrypt it with a different key, otherwise you may also harm the security of CBC mode. If you have to use your current key then you should use the resulting ciphertext as IV instead of the plaintext input. However, for a random IV that's completely unnecessary: you can just store it next to the ciphertext without protection.
Supplemental: message integrity & authenticity
In general it is always a good idea to at least verify integrity and authenticity of the stored messages before using it. Using a mode of operation such as GCM or EAX should be preferred.
Note that neither GCM nor EAX need an unpredictable IV; the IV just needs to be unique - there are no other requirements apart from size; using a fully random IV/nonce of 128 bit is of course fine as well. They automatically will include the IV into the calculation of the authentication tag.
Using a HMAC over the ciphertext (also known as encrypt-then-MAC) has been proven very successful to provide integrity / authenticity as well but please note that you should include the IV in the calculation* if you decide to go with HMAC.