Could elevated liver enzymes be a sign of liver cancer?

While it could. Enzymes (alt/ast) indicate liver cell injury/inflammation: hepatitis - from viruses to drugs to alcohol. Alk phos may point to duct obstruction. But these can all be up or none be up in cancer traveled to the liver.
No. Liver enzymes may be elevated for several reasons - but is almost never a sign of liver cancer. Elevated enzymes occur when the liver is damaged by alcohol, hepatitis, diabetes, fat, etc. It is important to note that damaged/cirrhotic livers are at greater risk to develop cancer. However, in this setting the elevated enzymes reflect the pre existing liver damage, not the development of cancer.