Is there a relationship between density and viscosity of a fluid?
They are not directly related as such. Density is defined as mass/volume units $\ce{kg/m^3}$. Water has a value 1000, (liquid) iron 7000, (liquid) lithium 516 , ethanol 789 and carbon tetrachloride 1632 all in units of $\ce{kg/m^3}$.

Viscosity is the rate of transfer of momentum exerted on a plane of fluid when a second plane of fluid a parallel distance $x$ away is moved with viscosity $v$.  The force exerted is $F=\eta \frac{v}{x}$, where $\eta$ is the viscosity.

The fundamental process that takes place when a liquid changes shape is that one layer of it has to slide past another. If we suppose that the liquid could be divided up into cells, then one cell has to 'hop' past another so that the liquid can move. The more difficult that this is the greater the viscosity. Thus to 'hop' a cell has to have an activation energy, that is it has to accumulate in itself enough energy to overcome resistance from other cells that prevent it from moving. Now defining just what a cell is is somewhat vague, but it is larger than a single molecule. But leaving that aside, the activation energy does reflect the inter-molecular interaction and shapes of individual molecules. Spheres with little inter-molecular interaction, e.g. small van-der-Waals forces, no hydrogen bonds, will have a lower viscosity than an awkward shaped molecule such as glycerol with the ability to form lots of hydrogen bonds.

Viscosity has units $\ce{Pa~s = N~m^{-2}S^{-1} = kg~m^{-1}~s^{-1}}$ but more commonly centipoise where $\ce{1~cP = 10^{-3} Pa~S}$. Some values  (in CP at 25 $\ce{^oC}$) are water 1, $\ce{CCl4}$ 0.97, iodoethane 0.56,  ethanol 1.07, cyclohexane 0.89, cyclohexanol 57.5, 1-propanol 1.945, ethylene glycol 16, glycerol 934.