Do I need to pull up the I2C voltage to the VCC of a ADC?
Is it safe to pull up the SDA &amp; SCL lines of the ADC to 1.8V when the ADC VCC is 3.7V?


Assuming you use pull-up resistors in the range of k&Omega;, the chance of a fire hazard or electrical shock is very small.
EDIT However, when data from this device in this configuration is used in a safety application, it could form a safety hazard, because the I2C is driven in undefined states as elaborated below.  

Will it work?

No, it will not.
According the datasheet VIH must be at least 0.7 * VCC.




  The -main bus voltage that supplies the LTC2451 &amp; the MCU is 3.7V


So, in your case Vcc = 3.7 V. And therefore, VIH > 0.7 * 3.7V = 2.6 V.
Pulling the lines up to 1.8V will not achieve that.

You should check whether the MCU allows for voltages higher than 1.8V on the I2C bus, otherwise you'll need a level shifter. 

If you read the datasheet further (page 7), from the point of view of the LTC2451 only the SDA requires a pull-up to VCC (because the device can only be in slave mode).