There are beahvioral, physiological, and psychological indicators of happiness, and on the flipside of depression, all of which are quantitative.
Levels of the neurotransmitters serotonin and dopamine in the brain are a physiological indicator of depression. Higher levels of serotonin are correlated with increased happiness and decreased anxiety. Dopamine also contributes to good mood, and in addition causes an increase in activity and willpower. Both serotonin and dopamine show decreased levels in depression. However, it is important not just to look at general levels of the neurotransmitter, but to also consider which parts of the brain show decreased neurotransmitter levels. Depression correlates with decreased serotonin and dopamine levels in the amygdala, thalamus, and hyppocampus. (http://www.health.harvard.edu/newsweek/what-causes-depression.htm)
Behavioral tests of depression are used on laboratory animals. The most common tests are the forced swim test (in which scientists measure how long a mouse persists swimming in a water container before it gives up) and the tail suspension test (in which scientists measure how long a mouse struggles to free itself when suspended by its tail). (http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3270071/)
In addition, self report questionnaires are designed to quantify depression in people. Some popular ones are the Beck Depression Inventory (BDI-II) (http://en.wikipedia.org/wiki/Beck_Depression_Inventory) and the Beck Hopelessness scale (BHS) (http://en.wikipedia.org/wiki/Beck_Hopelessness_Scale).