I had fractional laser last month on my face, will the grid marks + redness associated with a brown discolouration ever fade?

This is difficult to answer without pictures. It can take several weeks for laser results to settle. I  recommend following up with your Doctor about after care. I recommend skin care with vitamin c, b5 hydrator, Sculptra, silicone based scar gels with growth factors and enzymatic washes with papaya enzyme. PDT therapy can help calm inflammation. Typically I recommend adding  clear and brilliant, viva 4-6 weeks after laser along with microneedling/prp to improve texture and tone. See an expert. Best, Dr. Emer

Hyperpigmentation can occur after almost any laser procedure. Topical treatments are often recommended, but must be used for an extended period to achieve a positive outcome. In some cases, light and laser treatments can also be used to address the hyperpigmentation. Make sure you are working with an experienced, board-certified dermatologist who can customize your treatment plan to your precise needs and skin tone. 

I generally treat post inflammatory hyperpigmentation with topical creams.  It can take several months of twice a day treatment of hydroquinone before the pigment will resolve, and sun protection is also important, as it can bring out the pigment that you are trying to suppress with the hydroquinone.  You can also try light LED light threatment (Omnilux is one type).  
Dr. Sykes